Medical Image Segmentation: Beyond Level Sets (Ismail’s part) 1

Basics of Level Sets (Ismail) 14

Active Curves 1

1 S S

Gradient Descent (1) 2 Functional derivative Regional terms of the form

Gradient Descent (2) 3  E S E S -  S = tt  E S E S

Gradient Descent (2) 3  E S E S -  E S E S -  S = tt  E S E S

Gradient Descent (2) 3  E S E S -  E S E S -  S = tt  E S E S

Standard boundary terms: Geodesic Active Contours 4 e.g., Caselles et al., 97

5 Standard boundary terms: General derivative with E-L equations

5 1 Boundary length Standard boundary terms: General derivative with E-L equations

5 Standard boundary terms: General derivative with E-L equations Depends on image gradient Attracts curve to strong edges

5 Standard boundary terms: General derivative with E-L equations Depends on image gradient Attracts curve to strong edges

6 Standard region terms: Piecewise constant case e.g., Chan and et Vese, 01

6 Standard region terms: Piecewise constant case e.g., Chan and et Vese, 01 Alternate minimization (1) Fix parameters and evolve the curve (2) Fix curve, optimize w.r.t parameters

7 Standard region terms: Log-Likelihood

7 Standard region terms: Log-Likelihood Distributions fixed by prior learning e.g., Paragios and Dercihe, 02

7 Standard region terms: Log-Likelihood Distributions updated iteratively e.g., Gaussian: Rousson and Deriche 02 Gamma: Ben Ayed et al., 05

Functional derivatives for region terms (E-L equations and Green’s theorem) 8 See Zhu and Yuille, 96 Mitiche and Ben Ayed, 11

8 Curve flow in the log-likelihood case S Functional derivatives for region terms (E-L equations and Green’s theorem) See Zhu and Yuille, 96 Mitiche and Ben Ayed, 11

8 >0 Curve flow in the log-likelihood case S Functional derivatives for region terms (E-L equations and Green’s theorem)

8 <0 Curve flow in the log-likelihood case S Functional derivatives for region terms (E-L equations and Green’s theorem)

Level set representation of the curve 9

9 We can replace everything

Level set representation of the curve 9 Easy to show from the facts that on the curve: See Mitiche and Ben Ayed, 11

Alternatively, we can embed the level set function in the energy directly 10 Region terms: e.g., Chan and Vese, 01 Li et al., 2005

Alternatively, we can embed the level set function in the energy directly 10 Length term: e.g., Chan and Vese, 01 Li et al., 2005

Alternatively, we can embed the level set function in the energy directly 10 e.g., Chan and Vese, 01 Li et al., 2005 Region terms: Length term: Compute E-L equations directly w.r.t the level set function

Pros of level sets (1) 11 Applicable to any differentiable functional:

Pros of level sets (1) 11 Applicable to any differentiable functional:

Pros of level sets (2) 12 Direct extension to higher dimensions

Small moves + Fixed and small time step + Can be slow in practice: Cons of level sets (1) 13  S = tt  E S E S Courant-Friedrichs-Lewy (CFL) conditions for evolution stability  t < cst See, for example, Estellers et al., IEEE TIP 12

Sometimes very weak local optima Cons of level sets (2) 14 = 0  E S E S

Dependence on the choice of an approximate numerical scheme (for stable evolution) Cons of level sets (3) 15  e.g., Complex upwind schemes for PDE discretization See, for example, Sethian 99 See, for example, S. Osher and R. Fedkiw 2002  Keep a distance function by ad hoc re-initialization procedures